Number Theory: Basic to Advanced

Where the subject actually begins

Number theory has a reputation for being approachable, and it is: the questions can be stated with school arithmetic. The methods are not approachable in the same way, and the gap between understanding a question and being able to start it is where most students stop.

This course runs from divisibility and the Euclidean algorithm to quadratic reciprocity and Diophantine equations, in the order the mathematics requires rather than in order of how interesting each topic sounds.

Modular arithmetic is an algebraic system, not a remainder

This is the single most common blockage and it is worth naming.

A student who first met "mod n" as a clock analogy can compute remainders and cannot reason with them. They do not know when they may divide both sides of a congruence, why that question even arises, or what makes a linear congruence unsolvable. So when Fermat's little theorem arrives, they can state it and cannot use it.

We teach congruence as arithmetic in its own right from the beginning — with its own rules, its own failures, and its own structure. Everything from the Chinese Remainder Theorem onward depends on that foundation being real.

The three stages

Basic covers divisibility, the division algorithm, gcd and lcm, the Euclidean algorithm and its extended form, Bezout's identity, primes and the fundamental theorem of arithmetic, the sieve, divisor functions, and the introduction to congruences. These are worked until automatic.

Intermediate covers the Chinese Remainder Theorem, Fermat's little theorem, Euler's totient and theorem, Wilson's theorem, orders and primitive roots, and the arithmetic functions including Mobius inversion. This is the working core of the subject.

Advanced covers quadratic residues, the Legendre symbol, Euler's criterion, Gauss's lemma and quadratic reciprocity; Diophantine equations including Pythagorean triples and Pell's equation; descent arguments; and p-adic valuation with Legendre's formula and lifting the exponent.

The unit that decides everything after it

Orders and primitive roots is reliably the hardest unit and the most valuable. It is where a student has to hold several ideas simultaneously — the order of an element, its relationship to the group structure, and which moduli admit a primitive root at all.

It is also the unit that makes competition problems about exponents tractable. A student without it stares at a problem about the last digits of a tower of powers; a student with it knows exactly where to begin. We give it the time it needs rather than hurrying on to the more glamorous reciprocity material.

Deciding solvability without solving

Quadratic reciprocity is where the subject starts to feel like something deeper than clever manipulation. The Legendre symbol and Euler's criterion let you decide whether a quadratic congruence has a solution without producing one, and reciprocity turns that decision into a short computation.

Students find this genuinely surprising the first time, which is a good sign: it means they have understood what is being claimed.

Proof, from the first week

Number theory is a proof subject. An answer without an argument earns very little in RMO, INMO, the Waterloo full-solution contests or any undergraduate examination, and the habit of writing one does not appear by itself.

Written work is marked on whether a reader can follow it: assumptions stated, the awkward case handled rather than the convenient one, the conclusion actually reached. Most students find the first month of this uncomfortable. By about week six they are pre-empting the objections themselves.

Who this suits

Students from Class 9 upward through undergraduate study, with school algebra and no prior number theory. It suits students preparing for IOQM, RMO, AMC, the Integral Cup or any olympiad pathway, and equally students who want the subject for itself.

Classes are live and online, in groups of at most eight, with written work corrected individually.